Chapter 9: On the Species of Third Order Lines

Summary: Euler classifies third-order lines into exactly 16 species using only the criterion of branches at infinity, with detailed case-by-case derivation and a side-by-side comparison to Newton’s 72 species.

Sources: chapter9 (§§219–238).

Last updated: 2026-04-30.

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Program

§219 declares that “the nature and number of branches going to infinity are quite properly considered to be the essential criteria for assigning a curve to its species.” Second-order lines have already been so classified (ellipse / hyperbola / parabola — see classification-of-conics). The same method is now to be applied to third-order lines.

§§220–222 recapitulate the second-order classification through this lens, as a warm-up:

• §§220–221 — for $\alpha y^2+\beta yx+\gamma x^2=(ay-bx)(cy-dx)$ with distinct real factors, each factor refines (after a rotation) to an asymptote of the form $u=A/t$. Two such factors give the hyperbola.
• §222 — for $\alpha y^2+\beta yx+\gamma x^2=(ay-bx{)}^2$, the equation collapses (after rotation) to $u^2+(\delta b+ϵa)t/g^3=0$, a parabolic asymptote $u^2=At$. Or, if $\delta b+ϵa=0$, it factors into two parallel straight lines.

“Thus we would have found all of the species of second order lines, even if we had not already discovered them.”

Cubic setup

§223 — the general cubic equation:

$\alpha y^3+\beta y^{2}x+\gamma y{x}^2+\delta x^3+ϵy^2+\zeta yx+\eta x^2+\theta y+\iota x+\kappa =0.$

The principal member $\alpha y^3+\beta y^{2}x+\gamma y{x}^2+\delta x^3$ has odd degree, so it has either one real linear factor or three. After a rotation that aligns the chosen factor with $u$, the equation takes the working form

$\alpha t^{2}u+\beta t{u}^2+\gamma u^3+\delta t^2+ϵtu+\zeta u^2+\eta t+\theta u+\iota =0,$

with principal member $u(\alpha t^2+\beta tu+\gamma u^2)$.

Four cases

The factor structure of $\alpha t^2+\beta tu+\gamma u^2$ (and any further coincidences) drives the enumeration:

• Case 1 (§224): only $u$ is real, ${\beta }^2<4\alpha \gamma$. Yields species I, II.
• Case 2 (§§225–227): all three factors real and distinct, ${\beta }^2>4\alpha \gamma$. Yields species III, IV, V (the originally tentative “1 of $A/t$ + 2 of $A/t^2$” combination is shown impossible, dropping the count from six to five).
• Case 3 (§§228–232): one factor doubled; the third-factor branch and the doubled-factor branches each split into sub-cases. Yields species VI through XIII.
• Case 4 (§§233–235): all three factors equal. Yields species XIV, XV, XVI.

Sixteen species in total — see cubic-species-classification for the full derivation and a tabulated asymptote signature for each.

Mapping to Newton

§§236–238 close the chapter:

• §236 — sixteen species exhaust all third-order curves; the seventy-two Newtonian species reduce to these. The discrepancy is not a contradiction: Newton additionally classified by bounded-region shape, which Euler ignores by design.
• §237 — for each species, Euler exhibits the simplest representative equation and lists which Newtonian species (numbered 1–72) fall under it; see simplest-cubic-equations.
• §238 — Euler proposes calling his 16 classes genera, Newton’s 72 species, and any further sub-divisions varieties. The proposal is offered as nomenclature only; the rest of Euler’s text retains the word species for the 16. See euler-vs-newton-cubic-species.

Related pages

• chapter-7-on-the-investigation-of-branches-which-go-to-infinity — branch-at-infinity criterion that grounds the whole approach.
• chapter-8-concerning-asymptotes — asymptote refinement machinery used case by case here.
• hyperbolic-and-parabolic-branches — §218’s $u^j=A/t^k$ vs $u^j=A{t}^k$ catalogue, whose entries are what Euler counts.
• classification-of-conics — the second-order analogue (3 species).
• cubic-species-classification
• simplest-cubic-equations
• euler-vs-newton-cubic-species